Abstract

The bacterial pathogen Listeria monocytogenes propels itself in the cytoplasm of the infected cells by forming a filamentous comet tail assembled by the polymerization of the cytoskeletal protein actin. Although a great deal is known about the molecular processes that lead to actin-based movement, most macroscale aspects of motion, including the nature of the trajectories traced out by the motile bacteria, are not well understood. Here, we present 2D trajectories of Listeria moving between a glass-slide and coverslip in a Xenopus frog egg extract motility assay. We observe that the bacteria move in a number of fascinating geometrical trajectories, including winding S curves, translating figure eights, small- and large-amplitude sine curves, serpentine shapes, circles, and a variety of spirals. We then develop a dynamic model that provides a unified description of these seemingly unrelated trajectories. A key ingredient of the model is a torque (not included in any microscopic models of which we are aware) that arises from the rotation of the propulsive force about the body axis of the bacterium. We show that a large variety of trajectories with a rich mathematical structure are obtained by varying the rate at which the propulsive force moves about the long axis. The trajectories of bacteria executing both steady and saltatory motion are found to be in excellent agreement with the predictions of our dynamic model. When the constraints that lead to planar motion are removed, our model predicts motion along regular helical trajectories, observed in recent experiments.

Polymerization of the cytoskeletal protein actin into a network of filaments is necessary for the motility of several infectious bacteria. These bacterial pathogens hijack the actin machinery of the host cell to form “comet tails” due to unidirectional actin assembly at one of their poles. The force generated by the actin network allows the bacteria to move within the infected cell and to other cells in it's neighborhood. The biochemistry of the tail formation process has been well studied in the case of the Gram-positive bacterium Listeria monocytogenes (1–3). In particular, it has been shown that only one bacterial surface factor, ActA, is required for the movement of Listeria in a medium containing a few other actin-related proteins from the cytoplasm of the host cell. This observation has led to in vitro motility assays in which polystyrene beads (4) or disks (5) and phospholipid vesicles (6, 7) coated with ActA are propelled by the comet tails formed by actin polymerization. Many of the proteins responsible for the movement of Listeria have also been found in the front end of a crawling cell, also referred to as the lamellopodium (1). Listeria is therefore a model system that has provided important molecular level insights on actin-based motility.

Although progress has been made at the molecular level, the connection between biochemical processes and force generation has not been fully elucidated. With the knowledge of polymerization kinetics at the single filament level, is it possible to predict propulsive forces/torques at the macroscale? Several models (8–10) have partly addressed this question by considering the relation between the force exerted by the actin network and the velocity of the cargo it propels. Measurements (11–13) of the force–velocity relationships have been reported for both Listeria and beads, but consensus has not been reached on a satisfactory theoretical model due to significant qualitative (and quantitative) differences between the measurements themselves. Furthermore, because most of the theoretical models consider motion along a straight line, they have not addressed the relationship between the torques and angular speeds of Listeria, which move in a variety of fascinating geometrical trajectories (Fig. 1). These shapes include small- and large-amplitude sine curves, serpentine shapes, translating figure eights, circles, and a variety of spirals.

Although trajectories can provide information on the propulsive forces and torques, the ability to control them is important in applications where targeted delivery of the cargo carried by the comet tail is desired. The only theoretical study (14) on bacterial trajectories to date considers the curvature of the comet tail resulting from random variations in the location of actin filaments pushing the bacterium. However, the inherent deterministic dynamical structure portrayed by the periodic trajectories of Listeria given in Fig. 1 has to our knowledge been overlooked. In this paper, we develop a dynamical model that provides a unified mathematical description of the trajectories shown in Fig. 1. Our analysis shows that the large array of 2D trajectories in Fig. 1 are predicted by a set of deterministic evolution equations that incorporate not only a propulsive force included in most of the microscopic models to date but also its rotation about the body axis. When the angular speed of rotation is varied, a rich array of trajectories with a well defined geometric structure is obtained. Upon removing the 2D constraint, our dynamical model predicts 3D helical trajectories observed in recent experiments (15). Furthermore, unlike microscopic models based on the physics at the level of individual filaments, we view the motion in terms of macroscopic variables, such as the total force and torque acting on the bacterium due to the action of all of the filaments in the network. Although we do not answer the mechanistic question of how the various molecules/filaments conspire to give very regular trajectories, the predictions of our macroscopic model provide a rich set of constraints that have to be satisfied by any realistic microscopic model.

Motility Assays

Motility assays were prepared by adding dead Listeria in Xenopus laevis cytoplasmic extract supplemented with actin and ATP-regeneration mix as described in previous work (4). After brief incubation on ice, 1.1 μl of the assay mixture was removed and squashed between a glass slide and 22-mm2 coverslip and sealed with paraffin. Once the slides were prepared, they were incubated at room temperature for ≈10 min, after which the bacteria in the samples developed comet tails and started moving in the assay. Phase contrast images of the motile bacteria were recorded for a duration of 300–600 s at time intervals of 1.5 s. In the majority of the cases that we observed, the heads of the bacteria were aligned with the tangent to the trajectory as shown in Fig. 2, and, in a few cases, the heads were skewed relative to the tangent. In this work, we only consider the former case, leaving the more involved analysis of the latter case for the future. Snapshots of representative comet tail shapes are given in Fig. 1; we present the complete shapes of tracked trajectories after we develop a dynamical model for the trajectories.

Geometry of bacterial trajectories. (a) For constrained 2D motion, the orientation of the bacterium is denoted by θ, and the length of the arc between any two points on the trajectory is denoted by s. (b) Unconstrained 3D trajectories are denoted by the polar and azimuthal angles, θ and φ, respectively. In both cases, the orientation of the bacteria coincides with the tangent vector of the trajectory and the direction of the propulsive force, F, which is offset from the body axis by a distance d. The angle ψ denotes orientation of the location of net force F in the plane normal to the bacterial long axis, as for a bacterium moving between a glass slide and a coverslip (refer to a Inset).

Dynamical Model for 2D Motion

If the long axis of the bacterium is aligned with the trajectory, time evolution of its centroid and orientation are given by the equations of motion for the arc length s and the angle θ in Fig. 2, respectively. In general, the propulsive force generated by the filaments is balanced either by the viscous drag from the surrounding fluid or by the frictional/adhesive force due to breaking of the bonds that tether some of the actin filaments to the surface of the bacterium (8, 10, 16). However, for bacteria moving in the cytoplasmic extract with a viscosity of ≈0.01 Pa·s, the drag force is ≈10 fN, which is much smaller than the forces that tether the filament network to the bacterial surface, which are in the range of few hundred piconewtons to a few nanonewtons (8, 11–13). A very recent experiment (16) that considers the temperature dependence of the speed of motile bacteria finds an activated Arrhenius behavior consistent with the breaking of tethered bonds. In addition, branching of the filaments at the surface can also influence the force–velocity curve (9). To accommodate all these possible scenarios, we will assume a general force–velocity relation ṡ = v0(F), where the right-hand side can assume any functional form depending on the dominant physical mechanism limiting the bacterial speed.

A change in the orientation of the long axis of the bacterium requires a torque directed out of/into the plane in which it moves. Such a torque can arise if the moment of the surface force distribution about the centroid of the bacterium does not vanish and/or from the surface moments exerted by the filaments. In either case, the torque acting on the bacterium can be shown to be equal to the moment produced by force F offset by a distance d and oriented at an angle ψ with respect the long axis as shown in Fig. 2a. The relationship between the parameters d, ψ, and the net torque is given in supporting information (SI) Appendix 1. For any non-zero angle ψ, the net torque perpendicular to the long axis, τ⊥ = Fd, has components along the normal to the plane of motion (z direction) and in the plane of motion. For constrained 2D motion, the latter component of the torque is balanced by the opposing torques from the glass-slide and the coverslip. Using Ω(τ⊥) to denote the magnitude of the angular velocity of the bacterium due to the torque τ⊥ for unconstrained motion, the angular speed θ̇ for 2D motion can then be written by taking the z component of the angular velocity as θ̇ = Ω(τ⊥) cos ψ. To keep our discussion general, we do not specify the functional form of the Ω − τ⊥ relation because it would depend on the details of the physical mechanism opposing rotational motion.

In general, the time dependence of the net force (F) and torque (τ⊥) can be very complicated or even random, but the well defined geometric character of the periodic trajectories in Fig. 1 suggests that they can be described by deterministic evolution equations. To derive these equations, we first consider two simple shapes, namely, the circle and the small-amplitude sine. Whereas the curvature of the circle is a constant, the curvature of the sine can be positive, negative or zero, as shown in Fig. 3. Because the curvature κ = dθ/ds = θ̇/ṡ, it can also be written as Ω(τ⊥) cos ψ/v0(F). It then follows that the force and torque and the angle ψ should remain constant for motion in a circular path. However, at the points of inflection in the case of the sine curve, where the curvature of the trajectory vanishes, the angular component cos ψ should switch sign, because, by definition, Ω > 0 and the arc length should only increase with time, or ṡ = v0 > 0. A simple way in which this switch can be accomplished is if the point of application of the net force F moves in a circular path about the long axis of the bacterium, so that ψ = ωt, where ω is an angular speed. When the force is located to the left (right) of the pole (refer to Fig. 3), the torque is directed out of (into) the plane of motion, resulting in a positive (negative) curvature of the trajectory. On the other hand, when the force is above or below the pole, the z component of the torque, and hence the curvature, vanishes.

Trajectories of Listeria obtained by numerically integrating Eq. 1. The shape of the trajectories depends only on the ratio of the angular speeds, Ω/ω, indicated for each curve. For all of the cases, the initial velocity (taken to be 0.1 μm/s) is along the vertical direction, and the trajectories were tracked for a duration tmax = 1.75(2π/ω) (with ω = 0.03 rad/s). Note that the total distance traveled by the bacterium along each of the trajectories is v0tmax. For the sinusoidal curve, we have also shown the movement of the resultant force relative to the pole of the bacterium at four points along the trajectory.

In what follows, we first consider the situation in which the quantities v0, Ω, and ω are independent of time and demonstrate that, depending on their relative magnitudes, a large array of trajectories with a rich mathematical structure is obtained. We then show that the observed paths of Listeria correspond to one of these trajectories or can be obtained by applying small perturbations to these basic solutions. Because the effective force moves at a constant rate ω about the long axis, the angular velocity θ̇ and the orientation of the long axis of the bacterium can be expressed as
where θ0 is a constant of integration. One can also add an arbitrary phase to the argument of the cosine, but this can be set to zero without loss of generality. If at t = 0 the long axis is oriented along the y axis, θ0 = π/2, and the velocity of the centroid of the bacterium, (ẋ, ẏ) = v0[cos θ(t), sin θ(t)], becomes
This equation can now be integrated to obtain the bacterial trajectories [x(t), y(t)].

Bacterial Trajectories: Theory and Observations

The dependence of the trajectory shapes on the parameters v0, ω, and Ω can be obtained by rescaling the bacterial coordinates and time to render Eq. 1 dimensionless. If the length and time are rescaled by v0/ω and ω, respectively, the dimensionless velocities in Eq. 1 become functions of only the ratio Ω/ω. The shapes of the trajectories then depend only on this ratio: Altering v0 does not change the shape but only the overall size of the trajectories. Before considering the shapes for any arbitrary value of Ω/ω, it is instructive to consider two limits, Ω ≫ ω and Ω ≪ ω. In the former case, where the spinning rate of the force about the long axis is small, Eq. 1 becomes (ẋ, ẏ) = v0(−sin Ωt, cos Ωt), which corresponds to motion in a circle of radius v0/Ω. In the other limit, Ω ≪ ω, we get (ẋ, ẏ) = v0[−(Ω/ω) sinωt, 1], which then corresponds to motion along a sinusoidal curve with wavelength 2 πv0/ω and amplitude v0 Ω/ω2. The two limits give topologically distinct trajectories. In the former case, the motion is confined to a small region, whereas large spin rate in the latter case leads to delocalized motion along an infinitely long line that is nearly straight. How do the trajectories evolve from one of these paths to the other as the ratio Ω/ω is altered? As we show below, the nested sinusoidal functions in Eq. 1 lead to a sequence of geometric shape transitions that allow a smooth passage between these limiting cases.

For any arbitrary value of Ω/ω, Eq. 1 cannot be integrated to obtain closed-form expressions for the trajectories. We have therefore numerically solved these differential equations by using Mathematica and have plotted the trajectories in Fig. 3. The bacterial speed v0, the spinning rate ω, and the tracking time are identical for all trajectories. Although the trajectory shapes are distinct for different values of Ω/ω, they share several common features: (i) All of the trajectories are periodic with a time period of 2 π/ω; (ii) in each period, the curvature is positive for half the duration and negative for the other half, and therefore there are two points on each curve where the curvature vanishes; and (iii) the total length traveled by the bacterium along each path is identical. In line with the analysis from the previous paragraph, for small values of the ratio Ω/ω, sinusoidal trajectories are observed, but as this ratio increases the trajectories become more curved leading to the S curves, serpentine shapes, translating figure eights, and a series of knots at Ω/ω = 3. The increase in the maximum curvature with increasing Ω can be understood by noting that the instantaneous curvature of the trajectory is given by θ̇(t)/v0. The largest and smallest possible curvatures (in magnitude) are then Ω/v0 and zero, respectively.

In Fig. 3, we have only considered trajectories for 0 < Ω/ω < π, all of which are distinct from each other. We now show that for Ω/ω ≥ π, the trajectory shapes are a series of generalized spirals, where the backbone of the spiral is given by a basic shape in the region 0 < Ω/ω < π and the number of loops in the spiral is determined by the ratio Ω/πω. Examples of such trajectories are given in Fig. 4. To derive this result, consider the interval of time, π/ω, during which θ̇ in Eq. 2 and hence the curvature is either always negative or always positive. In this time interval, the change in the orientation of the bacterium can be found from Eq. 2 as Δθ = 2 Ω/ω. Because an increase in Δθ by 2π corresponds to the motion by the bacterium in a complete loop, the number of loops made by the bacterium in this time interval, after which the curvature of its path switches sign, is given by the integer closest to but smaller than Δθ/2 π or, equivalently, Ω/ωπ. A phase diagram that shows the number of loops as a function of the ratio Ω/ω is given in Fig. 5. In addition, near the points of inflection, where the curvatures are small, the shape of the trajectories is independent of the actual magnitude of Ω/ω. The trajectories therefore maintain their basic “backbone” given by one of the basic shapes in the regime 0 < Ω/ω < π.

Bacterial trajectories for Ω/ω = (Ω/ω)0 + nπ, where (Ω/ω)0 is 0 and 2 for a and b, respectively, and n = 1, 2, and 3. For n = 0, the corresponding trajectory is a straight line for case a and is given in Fig. 3 for case b. Note that when n increases by one, an additional loop appears in the trajectories. As in Fig. 3, we have chosen v0 = 0.1 μm/s and ω = 0.03 rad/s and have tracked the paths for tmax = 1.75(2π/ω); the total distance traveled along each of the trajectories is then v0tmax.

Normalized vertical distance (Eq. 3) traveled by the bacterium in the time interval 2π/ω (time to complete one full rotation about its long axis) plotted as a function of Ω/πω. When Ω/ω equals one of the zeros of the Bessel function, J0, the bacterium moves in closed orbits shown in blue. The number of loops made by the bacterium in the time interval between which the curvature of its trajectory switches sign is also indicated.

We now consider the limit Ω ≫ ω to see how the trajectories approach the circular path expected in this case. In the limit of interest, the number of loops in the trajectories becomes very large and several closely spaced circles appear near the point of the largest curvature. This behavior is evident even in the cases for which Ω/ω ≈ 10 in Fig. 4, where three nearly overlapping circles can be seen in the magnifications. The radii of these circles are approximately v0/Ω, in agreement with our earlier asymptotic analysis. The bacterium therefore spends a long time going around these circular loops as Ω/ω → ∞.

Another interesting observation one can make from Fig. 3 relates to the direction (up or down) in which the bacterium moves as a function of the ratio Ω/ω. Although initial velocities for all of the trajectories in this figure are directed vertically upwards, in four cases the bacterium continues to move up, and in two cases the trajectories are so curved that the bacterium eventually turns around and travels downwards. Furthermore, the distance traveled by the bacterium along the vertical direction in one period of the trajectory also depends on the ratio Ω/ω. To quantitatively determine this distance, Δy, we can integrate the y component of velocity in Eq. 1 to obtain
where J0 is the Bessel function of the first kind. The oscillatory nature of the Bessel function shown in Fig. 5 leads to the change in the direction of propagation seen in Fig. 3. Furthermore when Ω/ω equals one of the zeros of the Bessel function, net displacement vanishes, which means that the bacterium returns to its starting point. The trajectories in this case are closed orbits shown in Fig. 3. For the first zero of the Bessel function, the closed orbit is a figure eight, whereas other closed orbits have loops near the points of maximum curvature; the number of loops in the trajectories is given by the integer closest to but smaller than Ω/πω, as in the case of Fig. 4.

From a qualitative perspective, it is clear that all of the features of the comet tail shapes given in Fig. 1 can be reproduced with our dynamic model. We now turn to a quantitative analysis of bacterial trajectories with the aim of extracting the spin rate ω by fitting the observed trajectory shapes to the theoretical model. The shapes of several representative trajectories were obtained by extracting the coordinates of the centroids of motile bacteria and the orientation of their long axis from the time-lapse images by using subroutines from Matlab's image-processing tool box. The extracted trajectories are given in Fig. 6a–h (blue curves), and time-lapse movies of most of the trajectories are included as SI Movies 1–9. The red curves in Fig. 6a–h show the best fits to the theoretical model obtained by using a nonlinear curve-fitting program. The fitting parameters and the time of observation for each trajectory (tmax) are given in Table 1. We find that for observation periods ranging from 200 to 500 s, the dynamical model provides an excellent description of a variety of trajectory shapes, ranging from small-amplitude sine waves to spirals, as shown in Fig. 6a–f. The convincing agreement between observations and experiments shows that the forces and torques acting on the bacterial surface do not significantly change for ≈5–8 min for these trajectories.

Comparison between observed bacterial trajectories [blue and green for steady and saltatory motion (from ref. 17), respectively] and the best fits to the dynamical model (red). The fitting parameters for each of the curves is given in Table 1. (Scale bars, 5 μm.)

Two examples of longer trajectories recorded for ≈10 min are given in Fig. 6g and h. Although their overall features are in accord with the dynamical model, there are differences between these trajectories and the basic shapes in Figs. 3 and 4. In the case of Fig. 6g, the figure eights do not translate in a well defined direction but rather loop around each other, and the knots in Fig. 6h are not uniformly spaced as in Fig. 4a. We have therefore attempted to fit these shapes by adding a small-amplitude, low-frequency perturbation to the angular speed,
where α is an arbitrary phase. We find that excellent agreement with observations is obtained with the perturbation parameters given in Table. 1. Because the δΩ and δω are ≈5–10 times smaller than the unperturbed frequencies, we can conclude that the forces and torques due to the actin network are changing only very gradually over a time scale of 10 min, even in the case of these complex trajectories. We also note that in both of these cases the value of the dimensionless parameter Ω/ω is in a range where small changes in the kinematic parameters can lead to noticeable deviations in the trajectory shapes. This can be seen from Fig. 3, where significant changes in the shapes of trajectories are observed in the range 2.0 ≤ Ω/ω ≤ 3.0, compared with other cases, for example, 0.5 ≤ Ω/ω ≤ 1.5.

Several of the trajectories predicted by our dynamic model have also been observed in live Listeria. For example, very regular spirals with changing curvatures (similar to Fig. 6f) were observed by Gerbal et al. (8), and sine curves, winding S curves, spirals, and figure eights have been reported by Soo and Theriot (17). Because ActA, the bacterial factor responsible for motility, is present on the surface of dead bacteria, it is not surprising that the live and dead pathogens move in similar trajectories. Spiral trajectories also have been observed in the case of the nonpathogenic and non-actin-polymerizing bacteria Listeria innocua, which were conferred motility by expressing ActA on their surface (18). The recent work of Trichet et al. (19) shows that drops of oil pushed by actin comet tails also move along regular sinusoidal paths (refer to figure 1e of ref. 19).

Although we have focused attention on trajectories for bacteria moving at a steady speed, it is also well known that these pathogens also can execute saltatory motion, where the bacterium detaches itself from the actin tail for a brief period (typically a few seconds). During this period a new tail forms as a result of polymerization of actin filaments in 10–20 s, which in turn will be shed by the bacterium, so that the net result is a periodic motion referred to as saltatory motion. These type of trajectories are observed in wild-type bacteria (8, 17) as well as the mutant ActAΔ21–97 (8, 20). Typical trajectories of live wild-type bacteria showing this type of motion (from ref. 17) are given in Fig. 6i and j (green curves), and parameters for the best fits to these paths (Fig. 6i and j, red curves), calculated by using our dynamical model, are given in Table. 1. Note that the frequency at which the tails are shed and reformed at the bacterial surface (≈0.05 s−1) is much larger than the angular speeds ω and Ω. It is therefore not surprising that a macroscopic description in terms of the angular speeds is able to capture the overall shapes of the trajectories over a time period of 10–12 min. Furthermore, because a new network of filaments in the tail is formed after each saltatory period, close agreement between our model and the trajectories suggests that the parameters ω and Ω are determined by some characteristic property of the bacterial surface, perhaps the spatial distribution of the protein, ActA. This observation is further confirmed by SI Movie 9, in which a bacterium moving in a nearly circular trajectory is brought to a halt as it runs into cytoplasmic debris. In a few seconds, the bacterium is able to start moving as the propulsive forces due to polymerization overcome the resistance presented by the obstacle (also see SI Movie 6). Interestingly, the bacterium simply continues to move in the circular path as it did before it was completely stopped. We also note that the trajectory of the mutant bacteria (ActAΔ21–97) executing saltatory motion in figure 5A of ref. 20 and figure 8a of ref. 8 are perfect large- and small-amplitude sine curves, respectively.

Next, we consider the shapes of the bacterial trajectories in 3D, i.e., when the constraints that lead to planar motion are removed. We first consider the general case in which the force and torques acting on the surface of the bacterium are time-dependent, so that the velocities and angular speeds also depend on time. As in the case of constrained motion, the rate of rotation of the long axis is denoted by Ω(t), and ω(t) denotes the rate at which the point of application of the net force moves around the body axis. As we show in SI Appendix 1, whereas the closed form solution of the trajectories is difficult for any general form of Ω(t) and ω(t), the curvature, κ(t), and torsion, τ(t), of the trajectories can be expressed, respectively, as
This result shows that curvature of the trajectory is determined only by the angular motion of the long axis due to the offset of the net force from the axis, whereas the torsion is determined by the rotation of the point of force application around the long axis. When the translational and angular speeds v0, Ω, and ω are constant in time, the curvature and the torsion remain constant along the entire trajectory. We can then invoke the fundamental theorem for space curves (21) to show that the only space curve that satisfies this criterion is the circular helix. The radius and the pitch of the helix are related to the translational and angular speeds through R = v0 Ω/(Ω2 + ω2) and P = 2 πv0 ω/(Ω2 + ω2), respectively. In a recent experimental study, Zeile et al. (15) observed 3D trajectories of live Listeria and found that all of the bacteria show right-handed helical comet tails. For many of the trajectories presented in their paper, the pitch and radius of the helices appear to be constant for several helical periods (11 periods in figure 1a and six periods in figures 1j and 2 of ref. 15). In the case of the movie presented in the supporting material of ref. 15, the bacterium added a new helical repeat in ≈4 min, which corresponds to ω = 0.026 rad/s. This finding is consistent with the spinning rates inferred from the 2D trajectories in Table 1.

Trajectories for Gram-Positive vs. Gram-Negative Bacteria

It is important to note that, in our dynamical model, the periodic variations in curvatures were attributed to the motion of the net force around the long axis. In principle, the rate at which the bacteria spins about the major axis need not be the same as the angular rate for the motion of the net force. In this regard, there is a fundamental difference between Gram-positive and Gram-negative bacteria. Although rotation of Listeria about its body axis has been recently observed by Robbins and Theriot (22), they did not detect any body rotation during the actin-based movement of two Gram-negative bacteria (Yersenia pseudotuberculosis and Escherichia coli). However, Gram-negative bacteria move in 2D and 3D trajectories predicted by our dynamical model. Figure 3 of ref. 18 and figure 1a of ref. 23 show E. coli moving in 2D trajectories with nearly sinusoidal and figure-eight shapes, and figure 1b of ref. 24 shows periodic helical comet tails behind another Gram-negative bacterium Rickettsia rickettsii.

On the basis of our analysis, the torque perpendicular to the body axis, τ⊥leads to a change in the orientation of the long-axis bacterium. Clearly, this component of the torque is present in both Gram-positive and Gram-negative bacteria, but if the torque τ‖ aligned with the long axis can be transmitted to the bacteria from the network, they can also spin about their body axes. It is well known that ActA on the surface of Listeria cannot move relative to the body of the bacterium because it is rigidly anchored to the peptidoglycan cell wall. However, the fluid outer membrane of Gram-negative bacteria would allow the surface proteins like IcsA to move freely on the bacterial surface. Therefore, torque can be transmitted from the actin network to the bacterium in the former case but is dissipated by rotational diffusion in the fluid membrane in the latter case. We also note that, although the spinning rate of the bacterium reported in ref. 22 (0.01 rad/s) is comparable to the angular speed ω (refer to Table 1) at which the net-propulsive force moves about the long axis, the two rates need not be equal for individual bacteria. Future experiments on the correlations between the curvature of the trajectories and the movement of small beads attached to the bacterial surface (which can be used to measure the spinning rate) are necessary to provide information about the coupling between the two types of motion.

Concluding Observations

In conclusion, we have presented a unified dynamic model that explains the shapes of the actin comet tails of Listeria as well as other Gram-positive and Gram-negative bacteria moving in both 2D and 3D. In particular, we have shown that the large variety of 2D trajectory shapes that appear to be unrelated and complex are indeed predicted by a single set of dynamical equations based on the rotation of the point of application of the propulsive force about the body axis of the bacterium. These equations also predict motion along regular 3D helical trajectories when the constraints that lead to planar motion are eliminated. It is interesting to note that, although the actin comet tail is formed by continuous polymerization and depolymerization of a complex network of several hundred cross-linked filaments, the bacterial trajectories are described by only three kinematic parameters. These parameters are found to remain constant in time for 5–15 min, although the life-time of the individual filaments is only ≈1–2 min. Furthermore, the observation that the bacteria that run into obstacles or lose their tails during saltatory motion are able to resume moving along their original trajectories suggests that the kinematic parameters in our model are determined by a basic property of the bacterial surface that can vary from one individual to the other, for example, the surface distribution of the protein ActA. This observation brings us to the important question as to how the various molecules in the cytoplasm and on the bacterial surface conspire to actually give rise to the highly organized and regular macroscopic trajectories. As noted earlier, no microscopic theories to date account for or predict the rotational motion that leads to the periodic variation in the curvatures of the trajectories. We hope that the kinematic description of the macroscale trajectories derived in this work can provide guidelines for the development of a realistic microscopic theory.

From an experimental point of view, important insights can be gained by designing experiments that will allow the measurement of torque–angular velocity relationships for motile Gram-positive and Gram-negative bacteria and for beads coated with ActA or a similar protein, where the number of actin filaments and the degree of cross-linking in the tail can be varied in a controlled manner. The aspect ratio of the beads also can play an important role in determining the geometry of the trajectories, and a systematic experimental study of the trajectories of polystyrene beads as a function of their aspect ratio can provide valuable insights on actin-based motility.

Acknowledgments

We thank Rob Phillips for his hospitality and for his invitation to the physical biology boot camp in his laboratory at California Institute of Technology. This collaborative work was initiated by experiments performed during the boot camp. We thank Hernan Garcia, Mandar Inamdar, and Erik Klavins for help with the experiments, comments on the manuscript, and useful discussions. The work was supported by National Science Foundation Grants CMS-0210095 (to V.B.S. and D.T.T.) and DMR-0403997 (to A.P.).

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